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\title{Force$_{\text{2B}}$}
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\section{twobody interaction}
\label{sec-1}
A general two body interaction that is translation invariant, has a form  in coordinate representation: $V(\bm{r},\bm{r}')$ and in momentum representation: $V(\bm{k},\bm{k}')$, where all $\bm{r}$ and $\bm{k}$ variables are relative ones. In quantum physics, interaction is a operator that map a state to a state. Thus, interaction operator acts on a twobody state $\psi(\bm{r}')$ get a new twobody state $\psi(\bm{r})=\int{d\bm{r}' V(\bm{r},\bm{r}') \psi'(\bm{r}') }$. We have omitted all other freedom degrees like spin, isospin. If $V(\bm{r},\bm{r}')$ is in a form of $V(\bm{r})\delta(\bm{r}-\bm{r}')$, we say $V$ is local. Local interactions in momentum representation depend only on the transfer momentum $\bm{q}=\bm{k}'-\bm{k}$.
\subsection{Malfliet-Tjon interaction:}
\label{sec-1-1}
two Yukawa terms: one repulsive, one attractive.
$$V{(\bm{r})}=V_R \frac{\exp(-\mu_R r)}{r} + V_A \frac{\exp(-\mu_A r)}{r}$$
local: depent only on relative coordinate;\\
    spherical: indepent of direction of $\bm{r}$. \\
    parameters that reproduce nucleon-nucleon $^1S_0$ phase shift:
$$V_R=7.291\hbar c=1438.71\text{MeV~fm} \quad \mu_R=613.69/\hbar c=3.11\text{fm}^{-1}$$ \\
    $$V_A=-2.6047\hbar c=-513.98\text{MeV~fm} \quad \mu_A=305.86/\hbar c=1.55\text{fm}^{-1}$$ 
\subsubsection{solve the two body problem}
\label{sec-1-1-1}
\begin{enumerate}
\item In coordinate representation:
\label{sec-1-1-1-1}
radial equation is
$$(-\frac{\hbar^2}{2\mu} \nabla^2+V(r)+\frac{l(l+1)\hbar^2}{2\mu r^2} +V(r))u(r)=Eu(r)$$
where $\mu=\frac{m_N}{2}$ is reduced mass. The relative wave function $\psi(\bm{r})=\frac{u(r)}{r}Y_{lm}(\hat{\bm{r}})$.\\
     \emph{solve this differential equation by shooting method}\\
     \emph{solve in hamornic basis}
\item In momentum representation:
\label{sec-1-1-1-2}
$$\frac{k^2}{2\mu}u(k)+\int{dk' k'^2 V(k,k')u(k')}=Eu(k)$$
discretization the momentum by gauss-legendre: PHYSICAL REVIEW C 73, 034321 (2006)
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